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FUNDAMENTALS OF AHTEKNA RADIATION

By
hUGO amwm maps

A THESIS

Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Department of Electrical Engineering

1950

THESIS

 

II
in
IV

VI

VII

VIII

TAB LE OF C ONT LIT-TS

Section
Introduction
Qualitative Ideas
analysis
Experimental Results
Conclusion
Appendix I
Elementary Derivation of
Retarded Vector Potential
Retarded Scalar Potential
Appendix II
Tangential Induced Electric

Field of Dipole Antenna

Bibliography

Page

14
30
35

37
41

43
48

PREFACE

The purpose of this thesis is to present in a logical
and straightforward manner the fundamentals of antenna ra-
diation and related concepts which are not clearly explained
in some texts. The author wishes to eXpress his thanks to
Dr. R. D. Spence of the Department of Physics at M.S.C.,
whose criticism made the author self—critical of his own
ideas, to Mr. Stephen S. Atwood of the Department of Elec-
trical Engineering at the U. of M;,whose help and encourage-
ment were invaluable when even the dilemmas had dilemmas,
and to Dr. J. A. Strelzoff of the Department of Electrical
Engineering at M. S. C. for his help in the development.of

this thesis, and for his patience in reading the manuscript.

H. A. Myers

INTRODUCTION

It is a natural tendency to classify the fundamentals
of antenna radiation along with the fundamentals of D.C.
circuit analysis, operation of simple motors, etc., all of
which is pretty cut-and-dried material. However, the an-
swer to the question, "How does an antenna radiate?", is
quite difficult, and many authors and teachers who include
the study of antennas in their course material show a lack
of clarity on this point. It is one thing to arrive at the
equations for the radiation field and describe the radia-
tion from there, but quite another to describe how this
radiation is generated by the antenna. The crux of the di-
lemma is this: The tangential component of the electric field
intensity is necessarily practically zero at the surface
of the conductor because it is approximately zero inside,
and the tangential component is continuous across the boun-
dary. How then can there be any power radiated? Poynting's

vector relation is f z are, and if E = 0, then i = o, be-

cause H is certainly finite. Bronwell & Beaml

present the
conventional radiation-from-the-gap solution to this problem.
Their book is chosen as a Specific target because it is
used in our senior courses; however, they are not alone in

1 Bronwell, A. B., & Beam, R. E. Theory and Application of
Microwaves 1947 New York: McGraw—Hill

their lack of clarity on this point. Section 20.05 of
Bronwell & Beam* is quoted in its entirety so that the
reader may better grasp the problem of this thesis and so
that the author will not be accused of quoting out of
context.

"20.05. VALIDITY OF THE INDUCED-EMF- METHOD.-Although
the induced emf method of evaluating antenna impedances
yields results which agree favorably with measured values,
this method embodies certain inconsistencies which lead
us to question its validity. If we assume that the antenna
is a perfect conductor, then the tangential intensity
must be zero in order to satisfy the boundary conditions.
This presents an embarrassing situation, since if Ez is
zero at the conducting surface, then the normal component
of Poynting's vector is likewise zero and there can be no
power flow normal to the surface of the antenna. Does this
mean that a perfectly conducting antenna could not radiate
power? Experimental evidence indicates that the radiating
prOperties of an antenna are actually improved as the con-
ductivity of the antenna conductor increases. Intuitively
we would expect that a perfectly conducting antenna would
radiate Just as effectively as an antenna with finite con-
ductivity. If the radiated power does not leave the surface
of the conductor, then where does it leave the antenna?

Before attempting an eXplanation of the anomaly, - let
us straighten out the matter of current distribuition tithe
antenna. It is evident that, for a perfectly conducting an-
tenna, the assumption of a sinusoidal current distribution
is erroneous, since it yields a tangential intensity at the
surface of the antenna which we know cannot exist. However,
the value of E2 computed on the basis of an assumed sinusab-
dal current distribution is quite small and, for thin
antennas, only a slight modification of the current distri-
bution is necessary in order to cause E to vanish at the
surface of the conductor, thereby satisfyIng the boundary
conditions. Consequently, the current distribution along
a thin, perfectly conducting antenna would differ slightly
from a sinusoidal distribution, the discrepancy being most
pronounced at the current nodes. As the conductor diameter
approaches zero, the current distribution approaches a si-
nusoidal distribution.

* Ibid 1

"In order to answer the question "where does the radia-
ted power leave the antenna," we must search for a surface
over which the normal component of Poynting's vector is not
zero. Could this be the extreme end surfaces of the antenna?
In order to have a normal component of Poynting's vector, it
would be necessary to have a tangential component of electric
intensity at the end surfaces, but this is again ruled out
by our assumption of a perfectly conducting antenna.

"Let us now consider the an-
tenna-shown in Fig. 4, inwhich n
it is assumed that there is a fi-
nite separation distance between
conductors at the feed point ap.
An electric field exists in the
region g9 such that the line in-
tegral of electric intensity'over
this interval is equal to the im—
pressed emf. A magnetic field also
exists in this region owing to JJ
the current flow. Consequently
we would expect to find a nor- ( )
mal component of Poynting's vec-
tor over any surface enclosing
the region gp. This leads us to
suspected that the radiated power
might depart from the antenna in
the region between E and Q. As
the distance 9Q is decreased,the
electric intensity increases in
such a way that the line in-
tegral of electric intensityfTum
g to p always equals the applied
emf. In the foregoing deriva -
tions, a point generator was as- Fig. 4. - Field
sumed, implying that flxadistance in the vicinity
gp approaches zero. If this were of the feed point
true, however, the electric in- of a dipole anten-
tensity would approach infinite na.
value over the infinitesimal
distance ab.

"Since the induced-emf method apparently yields reason—
ably correct results, we conclude that the complex power
leaving a thin, perfectly conducting antenna may be deter-
mined by either of two methods.

1. Assume a sinusoidal current distribution andeapoint
source feeding the antenna. Ignoring boundary conditions,
evaluate Ez by the methods outlined above and insert this in
either Eq. (20.04-1 or 2) to obtain the complex power flow.

'2. With the true current distribution,Ez is zero at the
surface of the antenna,but it is not zero in the region ab.

 

 

:9

hr

 

 

U

-3—

We may therefore obtain the complex power flow by inte-
grating Poynting's vector over a surface enclosing the re-
gion ab.

It is not clear why the two methods give essentially
the same result. Apparantly the slight modification in an-
tenna current, which is necessary to satisfy the boundary
conditions, causes a shift from a situation of distributed
power flow from the conductor surface to one of concentrated
power flow emanating from the region.§p,without appreciably
altering the value of the power or the complex impedance
computed from it. Since all practical antennas have finite
conductivity, the value of E2. is small but need not be zero."

For many reasons which will be brought out in the text,
the theory that an antenna radiates from the gap is unsatis-
factory. Also, Stratton2 uses the e.m.f. method with no
such qualms and obtains thereby the correct value for the
radiation resistance. The following alternative eXplanation
will be presented.

For batteries, transmitting antennas - in general, for
generators of all kinds, as contrasted with passive
elements - it is necessary and logical to separate the net
electric field into two components: one is the charge-sepa-
rating or applied field, which is the electrical equivalent
of whatever force (mechanical, chemical, etc.) it is that
actually separates the charges, and the other is the induced
field, which is the usual electric field between thecxarges.
From this it will be demonstrated that it is logical that a
tangential component of the induced field exists at the

surfaces of the generator. Then it will be shown that the:

2Stratton, J. A. Electromagnetic Iheozy 1941
pp. 457 - 460. New York: McGraw-Hill

_ 4 -

Poynting vector theorem applies only to the induced compo-
nent of the net electric field, and from this it follows
that the power can be and is radiated from the elements of
an antenna.

The thesis will consist essentially of three parts.
First, in the QUALITATIVE IDEAS section, it will be ex-
plained in non-mathematical terms how as antenna actually
radiates. Second, in the ANALYSIS section, the mathematical
analysis will be presented. Finally, in EXPERIMENTS, the
eXperimental evidence will be presented that confirms the

mathematical and physical arguments given in the previous

sections.

QUALITATIVE IDEAS

There are many different ways to analyze antenna be-
havior. One is the theory of Schelkunoff3 which is based
fundamentally on the concept that the antenna and the Space
surrounding it are two wave guides. Maxwell's equations are
rigorously applied to a conical antenna, and then the usual
thin cylindrical antenna encountered in practice is approx-
imated by letting the angle of the conical apices approach
zero. However, analyses such as this are best appreciated
by those already familiar with antenna theory3the vieWpoint
to be presented is that of Hertz4, because this viewpoint,
at least in a qualitative sense, is more simple and straight-
forward.

In 1889 Heinrich Hertz*,after several years of research
on the nature and effects of electric oscillations and
radiation showed how the theory of Maxwell explained the
mechanism of radiation. Hertz applied Maxwell's equations to
an oscillating doublet and plotted the lines of electric
field intensity for various values of time. These curves are
shown in Figures 1 to 8 on the following page. The Maths -

matics involved is not very lengthy, but the actual plotting

3 Schelkunoff, S. A. Electromagnetic Waves pp. 441-479
1943 New York: D. Van Nostrand Co.

4 Hertz, Heinrich (translated by D. C. Jones) Electric
Waves 1900 New York Macmillan & Co.

* Ibid 4 pp. 137-150

 

MECHANISM OF RADIATION OF AN ObCIELAIINj DIPOLE

  
 
    

 

[ I, /
/
I I/ r" -,. ,
/ x/ // . r '
'7 fl II" .‘I/ r \ I‘- /
i f ,y /

   
  
  

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1"” ‘ . i‘ y _ ~ . ‘.
»- . i I ’ '- I
, \ 1 \ g f / x‘ K ,
'n ' , t ‘ - z
(' \. i 2; 3 \ , 5 ‘z i \ '-' .
‘ E i . i ' a y , t
”\x / ,l ,v r ‘I f K 5 1 f. ’ 1" 'y
\ Xi / 1" 1 ; fl " - \ I" 1] ~ I
/ i "I .' 1 '4 3; . / ’ l
k‘. ‘ ,l V, V ,, ‘K .t "t '\\’~”/ /' . '
.\~ . I V |\ \. ' /," X I
‘. / 1‘ ' . fl /
*wa,’ ; \\ Max/x
‘\ ‘ ‘ '\\¥
\ L.”- '

 

T — Period of complete oscillation

o — Positive Charge

0 — Negative Charge

 

of the curves is obviously a time-consuming job, because

there are so many of them. However, the important point is
that these curves are plotted from the results of Maxwell's
equations applied to an oscillating dipole, and therefore
they represent a true mathematical picture of the mechanism
of radiation. In the center of each figure the positive and

negative charges of the dipole are depicted in their approxi-

 

mate relative position at the instant of time for which the
figure is drawn. As Hertz mentions, the lines of force are
not continued right up to this picture, for the formulae
assume that the oscillator is infinitely short, and therefore
become inadequate in the neighborhood of a finite oscillator.
However, the infinitesimal oscillator that exists in practice
in the conductor of an antenna is the nucleus of the atom and

the associated free electron (or electrons).0r, in poor con-

 

ductors, the elementary oscillator may be thought of as an
atom with oscillating polarization. Poor conductors, then,
would not radiate as well as good conductors not only because
of increased 12R losses, but also because the excursion of
the electrons, and therefore the moment of the dipole (or
the current) would not be as great.

The curves of Figures 1 to 8 are drawn in color for a
complete cycle so that the mode of generation (of the black
lines of electric field intensity, for example) can be most
easily seen. The arrows indicate the direction of the field
intensity. Figure 1 shows the condition when the positive

-9-

and negative charges of the dipole are superimposed. In
Figure 2, one eighth of a cycle later, the electric field
lines begin to build up as shown, pushing fields which were
built up previously away from the source; this process
continues in Figure 3, where the separation of the charges is
a maximum and the current is zero. In figure 4, as the
charges begin to come together, the important part of the
radiation process begins to occur. The outer black fiehi
lines are prOpagating outward and eXpanding with the velocity
of light and are meanwhile in the process of being cut off
at their source. The red arrows indicate the direction of
the lines of force near the source. As the charges come to-
gether the lines of force must also, and since they are equal
and Oppositely directed (as the red arrows show), they
cancel in this region. The net electric field becomes zero
here, leaving the closed curves of field intensity shown
in Figure 5, where the current is now a maximum and Opposite-
ly directed to that in Figure 1. As the charges'begin.to
separate again in Figure 6 a new field builds up, Oppositely
directed to that in Figure 2, and pushes the black field
lines away from the source. Thus, radiation is effected.

The crude, symbolic pictures of atoms in various states
of polarization (or, atomic nuclei and the associated free
electrons) associated with each figure is the author's
attempt to show how the atoms of a radiating elemmt actually

serve as oscillating dipoles. The red rings represent the

- lO -

electrons, and the black spheres represent the nucleus and
all the shells except the outer one. Thus, by comparing
the statecfl'the atom with the position of the dipole charges
in each figure, it is easy to see how an atom can radiate
electromagnetic energy. Hertz writes:

"... This loss of energy corresponds to the radiation

into Space. In consequence of this the oscillation would of
necessity soon come to rest unless impressed forces restored
the lost energy at the origin. In heating the oscillation as
undamped, we have tacitly assumed the presence of such forces.
It remains now only to show how the atoms in the elements
of an antenna can be forced to oscillate in the manner
illustrated in Figs. 1 to 8, so that radiation will result.
Consider the half wave dipole antenna shown in Figure
9. At a particular instant of time the transmission line
feeding the antenna imposes positive charges on the gap

end of one element, while negative charges are imposed on
oo . lie
(3 3.1.0.8:311 8:833?
Fig. 9 1J hlé7l'“ transmission line

 

 

 

 

the gap end of the other element. As a result, the atoms are
polarized (or, the free electrons are attracted or repelled)
as shown. Near the center the positive charges attract the
electrons quite strongly, as shown. Then adjacent nuclei ate
tract the electrons of atoms further from the center less

strongly, and so on. This attraction (repulsion) effect

actually travels along the element with approximately the

- 11 -

velocity of light, and the result is that a standing wave
of current is established which is approximately sinus—
oidal, with the maximum at the center and the nodes at the
ends of the antenna. The charges at the end of transmission
line oscillate, of course, and thus force the atoms in the
elements to oscillate, and therefore radiate. The radiation
of the antenna is the summation of the radiation of its atoms.
The energy required to sustain the oscillations is an»-
plied by the transmission line, and as far as the transmis-
sion line is concerned, that energy is dissipated in a'uoa I
The load is represented as a resistance for computation pur-
poses, and this is, of course, the radiation resistance of
the antenna.

This is, then, the physical picture of the way in which
an antenna radiates. It is only qualitative, but certainly
any quantitative analysis should check the physical facts .
Specifically, an analysis which reveals that the radiation
can come only from the gap should be questioned, because
all physical reasoning and eXperience indicate that it is
the element that does the radiating.

G. H. Livens5 derives the equations for, and presents
the radiation curves of the Hertzian oscillator, and he
also examines the case and plots the curves taking into

account the damping which is really existent. In the Opinion

5 Livens, G. H. The Teory of Electricity pp. 292 - 313
1926 Cambridge University Press (2nd. ed.)

_ 12 -

of the author, these curves,if not the mathematics, should
be presented and explained early in the study of electro-

magnetic fields, so that radiation will be better understood.

-13-

ANALYSIS

The expression for the retarded vector potential A’ is

given as Eq. 16 in Appendix I as:

A : Tit-J )d (16)
fi/ffili : C v

An example of the use of the retarded vector potential A

 

is the derivation of the field of an incremental current
element (or antenna). The incremental antenna is shown in

Fig. 10, having a length g; and carrying a uniform current
IeJWt.

 

 

 

Fig. 10 - Coordinates for the incremental antenna

Applying Eq. (16) to a differential current element,replacing
Idv by EIdz, noticing that A'is in the same direction as the
current and is therefore kAz, the vector potential ata point

distant ; from the antenna‘ becomes:

AZ :qudz Jw(t-r/c) (22)
4Nr e

Since P ch, Eq. (22) can also be written

A =,uIdz 3(wt r)
z Afir e -9

-14..

Expressing the vector potential in spherical coordinates and

drOpping the time function ert, we obtain

Ar = AzcosG :,MIdzcos9e’J(5r (24)
4hr

A9 a AzsinG = juldzsinoe'Jfir (25)
Afir

A¢:O

The magnetic intensity is obtained by inserting Ar and
A9 into H = (lAu)curIA (in spherical coordinates). With the

additional substitution of (b: 2Tr/7( and remembering that

9/9¢ = O, we obtain Hr : HO : o
H¢ = J2fi'e ; IdzsinGe-jfir (26)

hr r2 4W'
To obtain the electric intensity, insert H¢ from Eq.

(26) into CurlH = ngE (Maxwell's second equation), giving

E : l - .Iggcosee'Jpr (27)
r “(r2 #3) 217

E = 2 l - x Idzsinee'fiwr (28)
e ”‘6? I r2 297:3)1777

The above derivation was taken from Bronwell & Beam*. The
following discussion is quoted from them.

"The electric and magnetic intensities contain terms
varying as l/r, l/r2, and l/r3. The components containing
l/r and l/r3 predominate in the immediate vicinity of the
antenna and are known as the induction field of the antenna.
The induction field represents reactive energy mnchis stored
in the field during one portion of the cycle and returned to
the source during a later portion of the cycle.The induction
field terms become vasishingly small at remote distances from
the antenna and hence do not contribute to the radiation of
power from the antenna.

* Ibid 1, pp. 402 - 405

_ 15 -

"The terms varying as l/r in the intensity expressions
comprise the radiation field of the antenna. The radiation
field is comprised of electromagnetic waves travel ng ra ially
outward with a prOpagation factor eJ(wt-@r): ey” t-r c and
with intensities which vary inversely as the first power of
the distance from the source. Equation (26) shows that the
induction and radiation field components of magnetic intensity
are equal at a distance of r = X/2Tf, or approximately a sixth
of a wavelength from the antenna.

It is interesting'EJObserve that if we had assumed that
the field buifidg up instantaneously t roughoutannce,‘i.g.,if
we had used e ‘9 instead of e3” t-rc inEq.(23), the radiation-
field terms would not have been present in the resulting inten-
sity equations.Radiation is therefore dependent upon'flmafact
that the field has a finite velocity of prOpagation.In conven-
tional circuit analysis it is customary to ignore the finite
velocity of prOpagation of the field. This approximation is
valid if most of the field is confined to a region which is
very small in comparison with the wavelength.It leads to what
is known as the quasi-statipnagy analysi .

"The radiation-field terms, taken alone, comprise a
Spherical TEM wave pr0pagating radially outward from the
source with a wave impedance equal to the intrinsic impedance
of free Space. Discarding the 1 factor in Eqs. (26) and (28),
we obtain the radiation field intensities,

 

H¢ : Iggsine e’JQ’r (29)

2hr ,

EG =IAI zsinGe'Je’r (30)
2Kr

The ratio of electric to magnetic intensity is equal to the
intrinsic impedance of the medium, thus

EG/H¢ : AA

"The power radiated by the incremental antenna is found
by integrating the normal component of Poynting's vector over
the surface of an imaginary Sphere having the antenna at its
center. For convenience we choose a Sphere which is large
enough so that the induction-field terms are negligible.
Inserting E9 from Eq. (30) into the expression for power

_ n 2
Pave —'iél
2N\

and dropping the phase-shift term e'59r, we obtain for the
time-average power density,

=E2= Id2120 2
m meg m

- 16 -

The total radiated power is therefore

“(T
P = { p2Tfr2sinG
o

= fl(Idz )2 2fir2sin3ede
8r A O

= (Id )2
13%: Z (33)

The radiated power is independent of the radius of the Sphere
over which the power density is integrated. This is a conse—
quence of assuming a lossless transmission medium."
Schelkunoff* arrives at equations identical to Eqs. (27)
and (28) by slightly different means and carries the process
one step farther by finding an expression for Ez on the axis
of the current element. Referring to Fig. 11: If the value
of Ez is to be found on the axis, then 9 = 0. Therefore E9

is equal to zero (r 75 O), and E2 equals Er evaluated at G = O.

=g211de(l + l/Jprb'Jflr (34)

Expanding 9'39? into its equivalent series, multiplying
through and colleting like terms gives,

EZ=MIdz(__ _3 4p 9;. m) (35)
Jar 2r

Schelkunoff then writes:

"The first two terms are in quadrature withI and on the
average do no work; but the third term is 1800 out of phase
with I and work is done against the field by the impressed
electromotive force. The in-phase component of this force is

then
Re(v1) = - dzRe(Ez) = 2dz21 = Qfifld§21 (36)
"8% 3).
The work done by this force per second is seen to be equal
to g in Eq. (33)." '

*Ibid 3. pp. 131 and 133-134
- 17 -

Thus,

P = VJ; =M (Idz)2
ave 2 3)?- (37)

which checks Eq. (33). The underlining of the word impressed
in the above discussion is the author's. The important point
is this: For each differential time-varying current element
there Mia component of its associated electric field E which
acts in a direction Opposite to the current direction. Thus,
an impressed or applied field must act in the direction of
the current do drive the current against the component of EZ
that is 1800 out of phase with I. Therefore, in a radiating
conductor the pp; field is

= E1 + E = 17¢ (38)

till

net

Since the conductivity gfis usually very large, Enet may be
practically zero, but this does not require that E be zero.

In fact, the above,deriVations Show that E’w111.not be zero.

E, as used above, is termed the induced field. The conceptof
the induced field is not an antenna phenomenon; it is used
throughout the whole range of Electrical Engineering, fron:

'D.C. batteries on, as will be shown later.In fact, the point
of this thesis is that many authors and teachers do not dis-
tinguish between the induced and the applied fields in anten-
na derivations, and as a result they run into the difficul-

ties explained in the Introduction.

- lg -

Poynting's theorem will now be examined critically.
Maxwell's equations are:

curl(E) = -‘o§/’at (I)

curl(H) 'i' +e'5/ot (II)
Form the scalar product of (I) by H and (II) by E and apply
the identity

div(ExH) = H'curl(E) -E°cur1(H)
to arrive at I

div(ExH) + SI = -E-(oB/ot) - fi-(eE/gt) (45)
Finally, integrating over a volume V bounded by a surface S,

and applying the divergence theorem Maths first term in (45):
f(ExH)-'fida +fE-Idv = -[(E-a'D'/’at + m/ot)dv (46)
S . V V

Stratton* is now quoted because his interpretation of the

E-Idv term in Eq. (46) is the same as the author's. Some
V ___ .
authors6 write that "... the third term (E'i) is the usual

ohmic term and so represents energy dissipated in heat per
unit time? This is incorrect,as will be seen from Stratton's

explanation.

“This result (46) was first derived by Poynting in 1884,
and again in the same year by Heaviside. Its customary interb
pretation.isas follows. We assume that the formal expressions
for densities of energy stored in the electromagnetic field
are the same asin the stationary regime. Then the right-hand
side of (46) represents the rate of decrease of electric and
magnetic energy stored within the volume. The loss of avail-
able stored energy must be accounted for by the terms on the

00......OOOOOOOOOOOOOOOCCOO00.0.0.0...OOOOOCOOOOOOOOOOOC0.0.

* Ibid 2, pp. 132, 133

6
Ramo, Simon, & Whinnery, J. R. Fields and Waves in Modern
Radio pp. 427 - 428 New York 1944 Wiley & Sons

- 19 -

left—hand side of (46). Let. Q? be the conductivity of the
medium and E' the intensity of impressed electromotive flxces
such as arise in a region of chemical activity - the interior
of a battery, for example. Then

izam.sq, §=Lo-r an
and hence .

‘j/E’Idv :‘f/(Ieflb)dv ij/vE'ofdv (48)
V V- V - .

The first term on the right of (48) represents the power.
dissipated in Joule heat - an irreversible transformation.
The second term expresses the power expended by the flow of
charge against the impressed forces, the negative Sign indi-
cating that these impressed forces are doing work pp the
system, offseting in part the Joule loss and tending to
increase the energy stored in the field. If, finally, all
material bodies in the field are absolutely rigid, thereby
excluding possible transformations of electromagnetic energy
into elastic energy of a stressed medium, the balance can
be maintained only by a flow of electromagnetic energy across
the surface bounding V. This, according to Poynting, is the
significance of the surface integral in (46). The diminution
of electromagnetic energy stored in'V is partly accounted
for by the Joule heat loss, partly compensated by' energy

introduced through impressed forces; the remainder flows out-
ward across the bounding surface S, representing a loss
measured in Joules per second, or watts, by the integral

[Fol-ids ef (am-3s. (49)
S S.

The Poynting vectoz; E defined by

E = ExE watts/metere, (50)
may be interpreted as the intensity of energy flow at as
point in the field; i.e., the energy per second crossing a

unit ar§a_whose normal is oriented in the direction of the
vector ExH."

By comparing E in Eq. (50) to Stratton's definition of
E in (47), it is seen that Poynting's vector theorem applies
only to the induced electric field intensity, E. This result
follows from the fact that Maxwell's Equations themselves in—

volve only the induced field E, as Strattons' development
implies. - 2O -

This will be illustrated by the Specific case of a d.c.

battery. See Fig. 11

{i

O O

 

 

4.

 

L.
i l
G)

g)

1'."
V
'0:

 

C99
6

 

 

 

 

 

71

 

4...... n
Fig. 11 Electric and magnetic fields of a d.c.battery.

It is common practice to consider only what happens outside
the battery, because an engineer is interested primarily in
circuit problems. However, it is rather widely known that
inside the battery there must be an impressed or applied
force which is in the direction of the current and which
drives the current against the resistance of the battery and
against the field due to the charges at the battery 'termi-
nals. This force is termed the impressed field E', and it
is simply the electrical equivalent of the chemical force
that separates the charges. Now, according to the usual
procedure, the power flow down the line is represented
as shown in Fig. 11. But notice which component of the net
electric field is used, It is the component that is direc-

ted downward from the positive to the negative charges, and

this component is the induced field! Since impressed fields
usually exist only in generators, the case of the d.c. bat-
tery illustrates the general fact that Poynting's theorem
applies only to the induced field.

Note that E', the impressed field, is in general not
strictly a field; it is usually only the electrical equiva-
lent of some force which exists inside a generator and not
outside. Thus, E generally can not satisfy the boundary
conditions imposed by Maxwell's equations because it can
have a tangential component on -the generator side of the
interface between the generator and the outside, but it is
zero everywhere outside. Therefore, Maxwell's equatiorls
can not be applied to E' i; general simply because sometimes
it is not a field but a force due to chemical, mechanical,
etc. action. On the other hand, the induced field E annsfies
Maxwell's equations for both the static and the time-varying
states.- Therefore, since E is the only component of the net
electric field intensity that can satisfy Maxwell's
equations, we conclude that Maxwell's equations in general
can be applied rigorously only to E, the induced field.
Since outside the generating source 'E' is usually zero, the
net field is equal to E, and here all authors are in agree-
ment as to the prOpagation of electric waves, reflection
phenomena, etc.

The reason for the qualifying "in general" andAhsually"

in the statements above is that in the Special case of a

_22-

transmitting antenna the field which is the driving force
for the antenna currents is itself the induced field of the
transmission line, and therefore Poynting's vector applies
to it. However, this field is closely associated with the
currents and exists only in the immediate vicinity of the
currents. When we are considering an antenna as a generator
we need not concern ourselves with hpg the antenna currents
are sustained, any more than we need to know the chemical
action by which the currents are sustained in a d.c.battery.
We are primarily interested in the power ppgpgpgpg from the
generator, whether this generator is a d. 0. battery, trans—
mitting antenna, or a 50,000 kw. alternator. To calculate
this generated power at any point in the field we must apply
Poynting's theorem to the induced electric field, the field
of the charges and currents, and p93 to the field or force
that drives the currents.

We are now in a position to examine how a differential
time-varying current element can radiate power. Eq. (4)
shows that the real component of the tangential induced
field intensity associated with a differential cureent ele-
ment is 180° out of phase with the current. It has been shown
above that the generated power can be calculated by applying
Poynting's vector to the induced component of the net elec-

tric field.intensity. Therefore, Poynting's vector, E = ExH,

must be directed as shown in Fig. 12.

- 23 -

-OI
U
ml
X
In

 

Fig. 12. Radiation of a differential

time-varying current

The current must be time—varying because Eq. (35) shows that
Re(Ez) is equal to a constant times the frequency squared.
Thus, it is seen that a differential time-varying current
element radiates power. Therefore, any conductor carrying a
time-varying current will radiate, because its current is
merely the summation of the differential current elements.
The radiation of a twin lead transmission line is negligible
only because the conductors are close together compared with
a wavelength; and since they are carrying currentin.0pposite
directions, the radiation of one is cancelled by the radiation
of the other.

Thus we are led to expect that the current-carrying
elements of an antenna must radiate electromagnetic energy.
If a good approximation to the actual current distribution
is found that can be integrated to give the associated elec-
tric and magnetic field intensities, we should expect that
Poynting's vector P will give the correct value for the

I - 24 -

power radiated by the antenna. King7 shows that a sinusoidal
distribution is good approximation to the actual current
distribution, and when the induced field associated with
this current is calculated it is found that is has a ppgl

component tangential to the conductop which is oppositely

directed to the current.From the above analysis, this result

 

should be SXpected. Furthermore, when the total ,generated
power is calculated by integnnnng E over the surface of the

conductors not only the correct value for the radiation
resistance is found, but also the correct value for the
antenna reactance is given!

In spite of these facts, almost every author except
Stratton writes that Ez Should be zero at the surface of the
conductor, and therefore no power can be radiated by the
= Le

is certainly almost zero,lnfl;E, the indpced field from which

[11”

conductors. The point they miss is that Enet = E' +

the radiated power pppp be calculated is definitely not zero.
These authors write that the EZ calculated from a sinusoidal
distribution is not zero because the sinusoidal distribution
is not sufficiently close to the actual current distribution.
The truth is that no matter what kind of a distribution is

assumed the real component of the tangential induced field
will not equal zero and will be Oppositely directed to the

7 King, Mimno, Wing Transmission Lines, Antennas, and wave
Guides pp. 90-93 New York 1945 McGraw—Hill

_ 25 -

current. Since the differential current elements have oppo-
sitely directed Re(Ez)s, if therezhsany current distribution
at all it will have a Re(EZ) 180° out of phase. Stratton*
writes,

"The method of Poynting, which has thus far been em-
ployed to calculate the radiation resistance, is based en-
tirely on a determination of the outward flow of energy at
great distances from the center of the radiating system.

Now if energy is constantly lost from the system it must
also be supplied by the source; consequently, work must be
done on the antenna currents at a constant rate. There must
be a component of the e.m.f. parallel to the wire at its
surface which is 1800 out of phase with the antenna currentf'

Serious difficulties are encountered as the resultcm
insisting that EZ be zero at the surface of the conductor.
It follows immediately, as Bronwell and Beam** Show, that
the radiation must come from the gap. This seems just barely
plausible for a half-wave dipole. But then, why does a full-
wave dipole radiate off at angles instead of perpendicular

to the gap? How does a delta—fed half-wave dipole manage to

radiate at all? It doesn't have a gap, as shown in Fig. 13.

 

Fig. 13 Delta-fed half-wave dipole

0.0.0.....0.0.0.0....OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO

* Ibid 2 p. 454
** Ibid 1 pp. 432 - 433.

- 25 _

In fact, why have elements on the dipole at all, if it is
the gap that does the radiating? Perhaps we need the current

so that a magnetic field will exist in the gap. Then how

‘ a" ........

I...
.0. ..‘
I

..........
0 §
3” —-> ‘~., 3" —b F‘.

. s '0' ~ 9‘ o '..
’0 '.... .II 0.. \ / ’0' .0. .0... '0'.
.’ = .' °'. ." ". .3
f II E ' : 3 a : 3
' ' g '. i ' 3. °,

 

 

Fig. 14. Current distribution on a theoretical
a colinear array.

do the elements of the colinear array shown in Fig. 5 radi -
ate? They have no gaps at the current maximums. Also, high
frequency transmission lines have a small amountcfl'radiation
loss, and there are certainly no gaps in the transmission
line wires. These questions alone, without the support of
Stratton or the mathematics presented in this thesis, were
sufficient to cause the author to be skeptical of the valid-
ity of the radiation-from-the-gap argument.

It is instructive to examine the lines of power flow
(Poynting vector) surrounding an antenna and its trans-
mission line feeder. See Fig. 15. In Fig. 15 (a) the usual
representation8 is given. This shows the power flowing down
the line and being dissipated in the resistive load, which

OQOOCOOOOOOOIOOOOOOOOOCOCOOOOOOOOO0..OOOOOOOOOOOOOOOOOOOOOC

8 See, for example, Attwood, s. 5. Electric and Magpetic
Fields 2nd. Ed. p. 123 1941 New York Wiley
& Sons.

_ 27 -

is represented by the red section at the end of the line.
The resistance of the transmission line wires is neglected .
In Fig. 15 (b) the power-flow lines are shown terminating on
the elements; of the antenna. These power lines would be
calculated from the impressed field, E', which is the field
of the transmission line.As shown previously, this impressed
field drives the antenna currents against the induced field
due to the currents.

It cannot be overemphasised that when we are consider -
ing a transmitting antenna as a generator of electromagnetic
radiation we must calculate the generated (radiated) power
by applying Poynting's vector to the induced field, Just as
is done in the case of a battery or any other generator.
Thus, the power radiated by the induced electric field is
shown in Fig. 15 (c).

If there are no losses in the antenna, E' and E are equal
in magnitude but Oppositely directed, the net tangential elec-
tric field intensity is zero, and there is no ppp power flow
from the antenna. _Bfiut the radiation fielgi still emanates frpg
the elementspecause_;p_;s coppenpategrfor by the egual and
Oppopite fielg of the transpiesionfl. Note that Fig. (15‘)
gives only a physical picture of the lines of power flow be-

cause Poynting's theorem can be applied rigorously only to

 

closed surfaces. See Fig. (19) and the discussion given in

the CONCLUSION section.

- 28 _

5
44L)
“Ha—é
(e)
<——«——»
Lines of power flow
dpe to induced field ._._,
(E ) of antenna charges
and currents.
<4 -.
L4

 

 

—,
L
——9 -d-:

)

._,
E
n, ___....__...

->
Lines of power flow in-
the vicinity of a loss-

less line and a load.

(a)

i

 

 

Fig. 15

 

 

pl

E'x E

 

 

 

 

_ 29 -

(b)

41111:

_;:)/' fl :
_. D»
,/
t"

 

 

Lines of power flow due
to impressed elecpric
field intensity (E9 winch
drives the antenna cur-
rents against their re-
active forces.

EXPERIMENTAL RESULTS

A simple qualitative eXperiment to determine whether
most of the radiation from an antenna comes from the air gap
was performed as follows: Two half-wave dipole antennas were
constructed with elements of half-inch c0pper tubing fifty
centimeters long. The supporting masts were constructed in
the form of a 1 so that the gaps were readily accesible. The
resonant frequency of the antennas was found to be 146 meg-
acycles, and a small oscillator was used to drive the trans-
mitting antenna. See Figs. 16 and 17. When the receiving an-
tenna was placed one wavelength Gmmroximately 2 meters )
away from the transmitting antenna, the maximum pickup on a
crystal-microammeter combination was 160 microamperes, or
about half scale on the 300 microampere range. 75 ohm'Udn
lead transmission line was used to match the antennas. Since
all walls were several wavelengths away, reflections were
reduced to a negligible value.

In the first run, the gap of the transmitting antenna
was left unshielded, and the match at the oscillator was
adjusted for maximum pickup. The transmitting antenna was then
rotated and readings on the meter at the receiving antenna
were taken for ten degree intervals. The results are plotted
in Fig. 18. One half of the pattern is slightly smaller than
the other due to the loss in the transmission line as it
wrapped around the mast of the transmitting antenna. This

difference is unavoidable in the simple set-up used, and it

-30—

is of no great importance anyway. The radiation pattern is
slightly fatter than it should be due to the nonlinearity
of the crystal-microammeter combination in the lower range,
but this is also unimportant.

In the second run, a coffee can (3%" high by 4%" dia. )
with the cover on tightly was used to Shield the gap. 5/8
inch holes were cut:n1the tOp and bottom to admit the dipole
elements, and the aulwas supported by polystyrene insulators
between itself and the elements. A one-inch hole was cut in
the side to admit the transmission line feeder. With the
same match at the oscillator as used in the previous case,
the run was repeated for the antenna with the shielded gap.
The results are shown as the slightly larger pattern in Fig.
17. The transmitting antenna was jiggled a couple of degrees
_hlthe process of shielding the gap, and this shows up in
the patterns. The effect of the shield can be seen because
the match to the oscillator was even slightly better with
the gap shielded than when the gap was unshielded. However,
this difference is only on the order of 5%, and merely shows
that the shield affects the match, as would be expected. The
important point is this: The radiatiop pattern of a dipgle
antenna with the gap shgelged gs essentially the same as the
pattern with the gap unshielded. The conclusion is obvious:
The radiation from an antenna does not come from the gap. If
it did, at least a large part of the radiation energy would

be dissipated in 12R losses in the shield, and the pattern

-31..

 

 

Fi
g. 17
Tra
nsmi
tt
ing antenna
with
shie
lded
83P-

- 32 -

 

'0

190:
1'.

y
. ‘g, .-

’
fl .
.
.
I
z

' i

_ a3.
"

’TEO
(it; '

_ — r
1 . . .
r n S. .
o o- n o o lel
- ' .0 IA I I9. .
~ — o u o o e u 0--
. c . l o o a v ow
’ — e n e e O I a.
_ -. u , C e I .w
_ . _
,I ’ it‘ll." IIIIIOII‘ DI IIIIOII‘I.
f. 1‘ .r_ . .
ad 3 u . .
’ .
... . ._ . *
o . .1. a ‘ H
-. .I nu 1' o n
O , I
.. _ .. 1
_ .. . .
_ . e
‘

 

a
.1
t

I:
O
n I l I 0 0-1-1.-

 

_
m o
4 .:.: _ .. ..
o m c

 

r. A

.
‘r
“g ". L’ ‘ o—n- "Q’I" v, 7“.

U.- V

‘ 23.71:
s

would be drastically affected.

Now certainly a small amount of energy is radiated by
the gap because there is a time-varying electric field in
the gap, which induces a time—varying magnetic field in the
vicinity, which induces another electric field in its vicini-
ty, and so on. But the above experiments show that at most
this is a relatively small amount of the total radiation of
the antenna.

A second experiment immediately suggested itself.Try
shielding the elements, since they are assumed to be the
source of the radiation energy. Thus, if the elements are
shielded, the radiation should decrease. Now this is not
a good experiment, because a shield around the elements
lowers the capacitance between them, alters the current dis-
tribution, and so on.

However, this experiment was performed in a qualitative
manner. Two sections of a cylindrical wave guide were used
as shields for the elements, and it was found, as eXpected,
that as the Shields covered more and more of the elementa
the radiation became less and less. The amount of power
decrease was roughly prOportional to the length of element
shielded. That is all that can and should be said about this

particular eXperiment.

- 34 -

CONCLUSION

The eXperiments in themselves do not present conclusive
evidence that most of the radiation of an antenna comes from
the elements, because it can argued that the _fields inside
the shield will leak through the necessary loles and induce
currents on the outside of the shield which would in turn
radiate. However, it seems probable that such currents would
introduce losses and/or distortions of the resultant field
pattern, and this was not found to be the case.

It must be emphasised that the comparison of a Hertzian
dipole to anv atomic nucleus and its associated free elec-
tron(s) in the presence of (an impressed time-varying electro-
magnetic field is not a rigorous one, because Maxwell's equa-
tion.can not be rigorously applied at atomic dimensiOnS.Nev—
ertheless, in the Opinion Of the author, this comparison
gives one a good picture of the physical mechanism of radia-
tion, and so it is presented here.

The important point of this thesis, however, is that the
generated power of a transmitting antenna must be calculated
by applying Poynting's theorem to the induced electric field
due to the currents and charges of the antenna. This concept
resolves the dilemma quoted from Bronwell & Beam in the
INTRODUCTION. Now Poynting's theorem can be rigorously ap-

plied only to closed surfaces*, and thisie Shown in Fig. 19.

* ... and only to electric and magnetic fields that are
functionally related, in the Opinion of the author.
however, this is a subject for a thesis in itself.

-35—

Fig. 19(a) Fig. 19(b)

J

”‘l‘

Pnet =J{égnetm)‘nds = O Pradiated ={féExH)-nds

In 19(a) the pp; power integrated over the closed surface S
is zero (neglecting losses), because the power flowing into
§ due to the transmission line just equals the power ra-
diated by the antenna. The radiated power must be calculated
from the induced field E, as in 19(b). Since E and E are

distributed over the total length of the antenna, it is rea-
sonable that the ppp;p antenna radiates. The vector F is
often interpreted as the intensity of power flow per unit
area, and Fig. 15 illustrates these flow lines. Also, Friis9
uses 15 as power flow per unit area in derivinga trensmissian
formula for antennas which has proven valid during years of
intensive use. Thus, since it has been shown that the phys-
ical theory, Poynting's theorem, and the eXperimental evi-
dence are in agreement, it seems reasonable that most of the

radiation of an antennggemanateplfrpp the elepents and not

from the air gap.

9 Friis, H. T. "A Note on a Simple Transmission Formula"
Proc. Inst. Radio Engg, Vol. 34, pp. 254 - 256-
May, 1946.

-35..

APPENDIX I
ELEHENTARY DERIVATION OF RETARDED VECTOR POTENTIAL

As a result of his eXperiments and derivation from.
Maxwell's equations on the prOpagation and reflection of
electromagnetic effects, Hertz* proved that these "electric
actions" were waves which propagated with a high but finite
velocity, this velocity being,ixlpoint of fact, the velocity
of light. The curves in the QUALITATIVE IDEAS section Show
how this is accomplished. Modern Radar makes Spectacular use
of this finite velocity of prOpagation (which will be desig-
nated by the letter 9, after common usage). The fact that
this is a finite velocity should appear logical even to the
layman, because the only alternative is that it be an gpigp-
gig velocity. But Modern Physics teaches that all electrmnqy-
netic waves have a mass associated with them, andinlinfinite
velocity would require an infinite amount of energy to
initiate it, winch is a physical impossibility.

At low frequencies the wavelengths are so long and the
velocity of prOpagation so high that all circuits are elec-
trically short, and the effect of the finite velocity is
negligible; but nevertheless, it exists. Ln general” from the
considerations of the previous paragraph, it can be said flmfl:

all electromagnetic action is prOpagated with a finite

* Ibid 4, pp. 107 - 136.

-37..

velocity. Thus, if the effect of a current (or charge) loca-
ted at the origin of a coordinate system is being measured at
a point P, which is at a distance 2 from the origin, P at
any Specific time p can measure only the effect of a current
which existed at some earlier time (t-r/c). The reason that
P cannot measure the current at time p is that the effect of
this current has not reached P. Therefore all of the standard
low frequency equations would be more accurate if their
currents or charges were written as functions of (t-r/c)
instead of merely as functions of p. This accuracy is, of
course, necessary at the higher frequencies.

Consider, for example, Ampere's law, which is illus-
trated in Fig. 20. All currents will henceforth be written
as I(t-r/c) to emphasise the fact of the finite velocity of

prOpagation.z

  

P(x,y,Z)

 

Ampere's law:

x dH = I(t—;/c)dln.n¢ (1)
Fig. 20 p r4r Tu

But (1) is only an equation of magnitudes.The vector form of

Ampere's law is: dR = 113:2écld1xrl (2)

1"

rl is the unit vector in the F direction

 

 

Or, (2) can be written dR = I(t-réc)d Ex; (3)
r
Integrating: E :‘JII(t-réc)de? (4)
r

_ 3g _

Consider now the portion of the integrand, dlxr.
pranding grad (l/r) in cylindrical coordinates gives

" d l ‘ = --
gra ( /r) _%3

Therefore, dlxg : grad(l/r)xdl (5)
r

Using the vector identity* Fxgradv : chrlF - cuerF,

simple algebra gives grad(l/r)xdl = curlgl - curldl (6)
r r

But since dl is only a directed line segment and can be ex-

pressed in general as (idx + jdy + kdz), simple expansion of

the curl gives curl(dl) : O. (7)

Therefore, d.x— = curl dl r 8)
7 </> <

Thus Eq. (4) reduces UJE :.r I(t-r/c)curl(dl/r). (9)

But I(t-r/c) is a scalar, and since we are dealing with
continuous functions, the order of differentiation and

integration can be interchanged. Eq. (9) then becomes:

 

 

h: : curl [flit-r/c)d-l-J (10)
r
But, by definition,
'13; 2 pH : curl}: (11)
Therefore, we define X :‘p “{I(t-r/c)dl (12)
r
But I(t+r/c) : i(t-r/c)od§ (13)

where da is the differential cross-section area
through which the current density I(t-r/c)flows.

* Ibid 3. p. 13, Eq. 8 - 4.

..39...

And [I(t—r/c)od§ di] : i(t-r/c)dv, (14)
where dv is the differential element of volume,

because i(t—r/c) is in the same direction as dl.

Therefore . X : lufff ift-rgcldv (15)
V r

The above derivation is patterned after that presented in
Ramo a flhinnery*, in which the unrationalized m.k.s. system
of units is used. when written in the rationalized m.k.s.

system of units, the expression for A becomes

K : Lerf-iLt-r/CMV (16)
4W V r

X is termed the retarded vector potential. The current den—

 

sity i is actually a function of x, y, z, and t - r/c.
Abraham & Beckerlo write the eXpression for the current as
i (x,y,z,t-r/c), but this is usually considered too cumber-

some, and so Eq. (16) is the formula usually found in the
11

,

literature.Most derivations, such as thecxuagiven in King
are too complicated for presentation to seniors.Ch.the other
hand, Brainerd, Koehler, Reich, & Noodruff12 get around the

problem by not defining K explicitly in the antenna deriva-
tions. However, the retarded vector potential K is an impor-

tant concept, and so its derivation is presented here.

* Ibid 6 pp. 73 - 74 and 162 - 164

10 Abraham, M., and Becker, T. Classical Electricity and Ma —
netism, pp. 220-222, 1932, New York, G. E. btechert

11 King, R. W. P. Electromagnetic Engineering pp. 224-232
1945 New York McGraw-Hill

12 Brainerd, J. 3., Koehler, G., Reich, J.J., & Woodruff, L.F.
Ultra-HighrFrequency Techniques pp. 392-394 1942
New York D. Van Nostrand Co.

- 4O -

RETARDED SCALhR PCTEBIIAL

The derivation given here is similar to the one pre-
sented by Skillinng. The potential due to a point charge
distance 3 away (taking into account the finite velocity of

prOpagation) is:

V = g(t-r[c) (l7)
4Trr

If the potential is due to many charges located at various
points in space, the potential is merely the sum of the

various potentials

n
V = _l__Zgj(t-r[c2 (18)

Finally, if the number of charges qi increases indefinitely
‘in a given Space so that a continuous volume distribution of

charge is approached, than

V : fijffirogt-r/cldv (19)

1. r

 

p is the charge density perunit volume and is a continuous
function. V is termed the retarded scalar potential.
Note that V is directly obtainable from the fundamental

Coulomb's law:

: 1‘1 20
4%%292 ( )

F is the force between the two charges. The electric field

'1”

intensity E :is defined as the force on a unit positive

Skilling, H. H. Fundamentals of Electric Waves pp. l27~
128 1942 New York John fiiley & Sons

_ 41 _

charge, so that E (taking into account the finite velocity
of prOpagation) is given by

E q(t-r[c)rl volts/meter (21)
4UEr‘

E is then integrated from infinity to,; to give Eq. (17).
Thus, the expressions for the retarded vector and scalar
potentials are easily derived from three fundamental facts
about electricity:(l) Electromagnetic effects are prOpagated
with a finite velocity: (2) Ampere's law: and (3) Coulomb's

law.

- 42 -

APPENDIX II
TANGEKTIAL INDUCED ELECTRIC FIELD OF DIPOLE.ANTENNA

A sinusoidal distribution is assumed for two reasons:
(1) For thin, cylindrical antennas, the sinusoidal distribu-
tion is a good approximation to the actual distribution. See
footnote 7, page 25. (2) The sine distribution is the only

one that can be integrated with any degree of simplicity.

i1

__l_

 

   
 

 

 

 

half-wave dipole
antenna with sinusoidal
current distribution

 

Fig. 21 ~k¢

The various parameters are illustrated in Fig. 21. The current
is given by

I = Iosinp(!/2 - 2) (51)
where the time function e3“t is assumed.

The charge distribution may be obtained by inserting'UKa

current from Eq. (51) into the equation of continuity

div(I) = --bqv/ct
qv 3‘9, the charge per unit volume. In one dimension

- 43 -

”DI/DZ = " 'Dq’l'(/’Dt : — jwqx
q! is charge per unit length. Differentiating (51) and sub-

stituting into the above equation gives

IO coss(R/2 - z) = - qu
q2 = — 310(2W[§)cos p(R/2 - Z) = -J;Ocos@((/2 - Z)
2fif Af
qg = -ilocost()/2 - Z) (52)
C

The retarded vector and scalar potentials are obtained

by inserting current and charge from Eqs. (51) and (52) into
(16) and (19).

 

 

I: = Lfi flt-r/cmv (15)
4“ V r
v = _1_ [Hf qv(t-r/c)dv (19)
4W9 Vol. r

For time functions of the form eJW(t-r/c),§

eJW(t'R/C) : ejwte-jwr/f :: ejwte’Jflr

Since quv = quZ and Idv : Idf, (16) and (19) become

(ejwt assumed)

+JV2 R j
: - — '- - I'd”
v 5%; -‘)2 jTogos@( /2 A)e fl a

after algebra

 

 

+ )2 jar
V : 'chfl ‘& cosB(9/2 - Z) e d2 (53)
4fi' - 2 r
’2
AZ 3 _l.0. + / sinBfl/2 - A) ejfirdz (54)
4n -«&2 r

The field intensities may be evaluated in terms of the

potentials by using Eqs. (11) and (43).

......OOOOOOOOOOCOOOO0.00.00.00.00...

* Stratton, Ibid 2 p. 455, uses e-jw(t-r/c)
only difference in our answers.

_ 44 -

and this is the

kl

(144)0ur1(X) (11)
E : - gradV -'OK/’at (43)
when expressed in cylindrical coordinates for this particular
problem, these become
Ez = -'DV/’DZ - JWAZ; 3p: “EV/”op; 33,; = o (55)
Hg; : -(l/M)DAZ/a(a (D6)
The intensity component EZ at p resulting from the
charge and current in the antenna is obtained by inserting

Eqs. (55) and (54) into the first of (55). Letting f(r)= e‘3 I;

n
we have: +1/ 2 +9/2 (57)
az. - J__I__ [f(r)cosfs(J/2 2] d2. -u rmsmm 1/2-z)dz
ETI— M -’/2’Dz -!/2
Referring to Fig. 17, we obtain
2 z ('92 + (Z -Z)2 (58)
”Dr/oz =-

nggfll); “or/DZ =1(_§;Z_)(-l)
Therefore ‘DIVbz : -ar;sz r

crud/92 = (or(r)/or)ay.$z; ’Df(r)/.)Z = (of(r)/or)or/oz
Therefore ’Df(r)/gz : -0f(r)/aZ

This result is now substituted into the first integral in Eq.

(57) and the integration is then completed by parts.

51/2
Ez(first integral) '— WC ”’2OfKr) /§Z°086(fi/2 ' Z)dZ
Set u = cosfi(9/2 -Z) v = f (r)
u : +psinQU/2 -Z)dZ. dv : (or(r)/sz)dz (59)

- 45 -

+9 +
-Q;: +W _£/:f(r)sinp()(/2 -Z)dZ

note thatMB: 420 ‘g = \( "é'WVMé: wu

when Eq. (59) is substituted into (57), the two integral

Ez(st.{) = —Mr(r)cos(3(9/2 -2)

 

terms cancel, leaving

“ 31 M '33? (t/2 z) +{/2
be - n e COS , '-
2 Eur -(/2

 

-£%9M[e-JPI‘2/r2 -<e-J%rl/ri>cos<bx] <60>

For a "g wave dipole (3) = (2 “AM/2 :Tr; cosYY : -1

£2. =-33010 (L'mr? + L39“) (61)
1‘2 1‘1

 

It must be noted that E can not be equal to zero in the in-
tegrals used in this derivation, because then the integrals
would be imprOper and would have to be evaluated by the meth-
ods of Advanced Calculus. Also, r1 and r2 can not be zero at
the ends of antenna, because then EZ would be infinite. All
these conditions are automatically taken care of in practice
because the elements of the antenna have a finite radius.

Eq. (35) shows that, for a differential current element,
it is the peg; component of EZ that is Oppositely directed
to the current. Thus, we would eXpect that the real component
of 32 in Eq. (61) would be Oppositely directed to the current,

since the current in any wire is merely the summation or inte-

gral of its differential currents. Expanding Eq. (61) gives

-45-

E : ~33OI — 1 (cos r - Jsin r ) + ;_ (cos r - jsin )
z o\'r2 (52 92 r1 451 W1]

Taking the real part:

Re(EZ) : -3OIO(sin§r2 + sin r1) (62)
- r2 r1

Thus we see that the real component of E is oppositely direc-

ted to the current as expected.

_ 47 -

10.

ll.

12.

13.

BIBLIOGRhPHY
Bronwell, A. B., & Beam, R.E., Theory and Application
of Microwaves, 1947 New York, NcGraw-Hill

Stratton, J. A., Electromagnetic Theory, 1941
New York, McGraw-Hill

 

Schelkunoff, S. A., Electromagnetic Waves, 1943
New York, D. Van Nostrand Co.

Hertz, Heinrich (translated by D. C. Jones), Electric
Waves, 1900, New York, Macmillan & co.

Livens, G. H., The Theory of Electricity, 1926
Cambridge University Press, Second Edition

Ramo, 6., & Whinnery, J. R., Fields and Waves in Modern
Radio, New York, 1944, Wiley & Sons

King, Mimno, Wing, Transmission Lines, Antennas, and
Wave Guides, New York, 1945, McGraw-Hill

Attwood, S. 5., Electric and Magnetic Fields, 2nd. Ed.
1941, New York, Wiley & Sons

Friis, H. T. "A Note on a Simple Transmission Formula"
Proc. Inst. Radio Eng., Vol. 34, May, 1946

Abraham, M., and Becker, R., Classical Electrgcity ang
Magnetism, 1932, New York, G. E. Stechert & Co.

King, R. W. F., Electromagnetic Engineering, 1945
New York, McGraw—Hill

Brainerd, J. G., Koehler, G., Reich, J. J., & Woodruff,
L. F., Ultra—High-Frequency Techniques, New York
1942, D. Van Nostrand Co.

Skilling, H. H., Fundamentals of Electric Waves, 1942
New York, John Wiley & Sons.

-48-

RCCM UM: uni;
_ - MY ‘8 '55